The Product Rule | 乘积法则

📚 The Product Rule | 乘积法则

The product rule is a fundamental differentiation rule used to find the derivative of the product of two functions. In A-Level Mathematics, mastering the product rule is essential for tackling calculus problems involving polynomials, exponentials, logarithms, and trigonometric functions. This article will guide you through the definition, derivation, examples, and common pitfalls.

乘积法则是求两个函数乘积导数的基本微分法则。在A-Level数学中，掌握乘积法则对解决涉及多项式、指数函数、对数和三角函数的微积分问题至关重要。本文将带你理解其定义、推导、示例和常见易错点。

1. What is the Product Rule? | 什么是乘积法则？

If we have two differentiable functions, u(x) and v(x), their product is u(x)v(x). The product rule states that the derivative of this product is given by:

若有两个可微函数 u(x) 和 v(x)，它们的乘积为 u(x)v(x)。乘积法则指出该乘积的导数由下式给出：

d/dx [u(x) v(x)] = u(x) v'(x) + v(x) u'(x)

This can also be written using prime notation as (uv)' = u'v + uv'.

也可用撇号记法写为 (uv)' = u'v + uv'。

In words, the derivative of a product is: the first function times the derivative of the second, plus the second function times the derivative of the first.

用语言描述：乘积的导数等于第一个函数乘以第二个函数的导数，再加上第二个函数乘以第一个函数的导数。

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2. Derivation from First Principles | 从第一原理推导

Although you are not required to reproduce the full derivation in an Edexcel exam, understanding it deepens your insight. Starting from the limit definition of the derivative:

虽然考试不要求完整重现推导过程，但理解它能加深认识。从导数的极限定义出发：

d/dx [f(x)g(x)] = lim(h→0) [f(x+h)g(x+h) − f(x)g(x)] / h

Add and subtract f(x+h)g(x) in the numerator, then split the limit to obtain f'(x)g(x) + f(x)g'(x).

在分子中加减 f(x+h)g(x)，然后拆分极限即可得到 f'(x)g(x) + f(x)g'(x)。

This shows the product rule is not an arbitrary trick but a direct consequence of the limit laws.

这表明乘积法则并非随意套用的技巧，而是极限定律的直接结果。

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3. Basic Examples | 基本示例

Consider y = x² sin x. Here we can set u = x² and v = sin x. Then u' = 2x, v' = cos x. Applying the product rule:

考虑 y = x² sin x。可令 u = x²，v = sin x。则 u' = 2x，v' = cos x。应用乘积法则：

dy/dx = (x²)(cos x) + (sin x)(2x) = x² cos x + 2x sin x

Always remember to differentiate each function separately and then combine the terms. Practising with simple polynomial-trigonometric products builds fluency.

始终记住分别对每个函数求导，再将各項结合。练习简单的多项式与三角函数的乘积有助于熟练运用。

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4. Product Rule with Exponential Functions | 指数函数的乘积法则

Exponential functions often appear in products. For example, y = 3x e²ˣ. Let u = 3x, v = e²ˣ. Then u' = 3, v' = 2e²ˣ (using the chain rule for e²ˣ).

指数函数经常出现在乘积中。例如 y = 3x e²ˣ。令 u = 3x，v = e²ˣ。则 u' = 3，v' = 2e²ˣ（e²ˣ 的导数需用链式法则）。

dy/dx = (3x)(2e²ˣ) + (e²ˣ)(3) = 6x e²ˣ + 3e²ˣ = 3e²ˣ (2x + 1)

Factorising the result is often a good idea, as it may help with stationary points or second derivatives.

通常最好对结果进行因式分解，因为这可能有助于求驻点或二阶导数。

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5. Product Rule with Trigonometric Functions | 三角函数的乘积法则

Trigonometric products require careful handling. Take y = eˣ cos 2x. Here u = eˣ, v = cos 2x. Then u' = eˣ, v' = −2 sin 2x.

三角函数的乘积需谨慎处理。以 y = eˣ cos 2x 为例。这里 u = eˣ，v = cos 2x。则 u' = eˣ，v' = −2 sin 2x。

dy/dx = eˣ (−2 sin 2x) + (cos 2x)(eˣ) = eˣ (cos 2x − 2 sin 2x)

The negative sign from differentiating cosine must not be forgotten. Always double-check the derivatives of trigonometric functions.

微分余弦时产生的负号绝不能忘记。务必仔细检查三角函数的导数。

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6. Product Rule with Natural Logarithms | 自然对数的乘积法则

d, u or v may be a logarithmic function. Let y = x³ ln(2x). Set u = x³, v = ln(2x). Then u' = 3x², v' = 1/x (since derivative of ln(2x) is 1/x).

当涉及 ln x 时，u 或 v 可能为对数函数。设 y = x³ ln(2x)。令 u = x³，v = ln(2x)。则 u' = 3x²，v' = 1/x（因为 ln(2x) 的导数为 1/x）。

dy/dx = x³(1/x) + ln(2x)(3x²) = x² + 3x² ln(2x) = x² (1 + 3 ln(2x))

Simplifying logarithmic products often reveals a clean factored form, making subsequent analysis easier.

化简含对数的乘积通常能得出简洁的因式分解形式，使后续分析更容易。

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7. Combining Product Rule and Chain Rule | 乘积法则与链式法则的结合

Many Edexcel problems require using both rules. Consider y = (2x − 1)³ √x+1 . Here u = (2x − 1)³ and v = (x+1)^1/2.

许多 Edexcel 试题需要同时使用两个法则。考虑 y = (2x − 1)³ √x+1。这里 u = (2x − 1)³，v = (x+1)^1/2。

We differentiate u using the chain rule: u' = 3(2x − 1)² × 2 = 6(2x − 1)². For v, v' = ½(x+1)^−1/2 × 1 = 1/(2√x+1).

我们用链式法则求 u 的导数：u' = 3(2x − 1)² × 2 = 6(2x − 1)²。对于 v，v' = ½(x+1)^−1/2 × 1 = 1/(2√x+1)。

dy/dx = (2x − 1)³ · 1/(2√x+1) + √x+1 · 6(2x − 1)²

Factor out common terms to tidy the answer. This layered approach is very common and worth practising thoroughly.

提取公因式以使答案整洁。这种分层式解法非常普遍，值得充分练习。

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8. Extending to Three Functions | 三个函数乘积的推广

an be extended to a product of three functions. For y = u(x) v(x) w(x), the derivative is:

乘积法则可以推广到三个函数的乘积。对于 y = u(x) v(x) w(x)，其导数为：

dy/dx = u' v w + u v' w + u v w'

Example: y = x eˣ sin x. Treat u=x, v=eˣ, w=sin x. Then u' = 1, v' = eˣ, w' = cos x.

示例：y = x eˣ sin x。视 u=x, v=eˣ, w=sin x。则 u' = 1, v' = eˣ, w' = cos x。

dy/dx = (1)(eˣ)(sin x) + (x)(eˣ)(sin x) + (x)(eˣ)(cos x) = eˣ (sin x + x sin x + x cos x)

This pattern follows from applying the two-function rule twice; remember to keep everything organised.

这一模式来自两次应用两个函数的乘积法则；记住保持各项条理清晰。

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9. Common Mistakes and How to Avoid Them | 常见错误及避免方法

One frequent error is omitting one of the two terms. Students sometimes write dy/dx = u'v only, forgetting the v'u term. Always write down u, v, u', v' explicitly.

一个常见错误是遗漏两项中的一项。学生有时只写 dy/dx = u'v，忘记了 v'u 这一项。务必明确写出 u、v、u'、v'。

Another pitfall is misidentifying which function is u and which is v. While order does not affect the result, it can make differentiation more complicated if not chosen wisely. Pick the function that simplifies upon differentiation as v, though this is a matter of taste.

另一个陷阱是对 u 和 v 的误判。虽然顺序不影响结果，但如果选择不当，可能使求导变得更复杂。通常可选择求导后更简单的函数作为 v，不过这取决于习惯。

Also, be careful with negative signs and chain rule within v' or u'. Write intermediate steps neatly to avoid sign errors.

此外，要注意 v' 或 u' 中的负号和链式法则。整洁地写出中间步骤以避免符号错误。

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10. Application in Optimisation Problems | 在优化问题中的应用

The product rule often appears in optimisation problems where you need to maximise or minimise a quantity expressed as a product of two variable expressions.

乘积法则常出现在优化问题中，此时需要最大化或最小化用两个变量表达式乘积表示的量。

rentiating using the product rule gives dA/dx, and setting it to zero finds the optimal x.

例如，一个矩形的一条边长为 x，另一边由函数如 (10 − ½x) 定义，面积 A = x(10 − ½x)。使用乘积法则求导得到 dA/dx，令其为零即可求得最优 x。

A = 10x − ½x² t a more complex product might be V = π x² (h − x) for a cylinder, requiring product rule.

更复杂的乘积如圆柱体积 V = π x² (h − x)，就需要乘积法则。

Thus, mastering the product rule unlocks the ability to solve real-world context problems that Edexcel loves to set.

因此，掌握乘积法则能解锁解决 Edexcel 青睐的实际情境问题的能力。

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11. Practice Problems | 练习题

Try these examples to test your understanding. Differentiate the following with respect to x:

尝试以下示例来检验理解程度。对下列关于 x 的表达式求导：

Function	Hint
(a) y = x² e³ˣ	Use u = x², v = e³ˣ
(b) y = ln x · cos x	Watch for the derivative of ln x
(c) y = (2x+1)⁴ (x²−3x)⁵	Chain rule + product rule
(d) y = sin 2x · tan x	Simplify tan x first if you like

Answers can be checked by expanding or using technology, but ensure you show full working. Practice builds speed and accuracy for the exam.

可通过展开或使用科技手段核对答案，但务必展示完整解题过程。练习能提升考试中的速度与准确性。

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12. Summary | 总结

e product rule is an essential tool in differentiation. You must memorise the formula (uv)' = u'v + uv' and learn to combine it confidently with the chain rule.

乘积法则是微分中不可或缺的工具。必须熟记公式 (uv)' = u'v + uv'，并学会将其与链式法则自信地结合使用。

Consistent practice, careful labelling of u and v, and tidy working will help you avoid common errors. With these skills, you will be well-prepared for any product rule question on the Edexcel A-Level Mathematics exam.

持续练习、仔细标注 u 和 v 以及保持演算整洁，有助于避免常见错误。掌握这些技能后，你将能从容应对 Edexcel A-Level 数学考试中任何乘积法则的题目。

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